Calculus II - Polar Coordinates

Introduction to Polar Coordinates

• Cartesian ((x,y)) vs. Polar ((r,\theta)) systems.
• Polar coordinates use distance from the origin (r) and angle (\theta) from the positive x-axis.
• Polar system allows (r) to be negative; negative (r) results in the point lying in the quadrant opposite (\theta).
• Infinite representations for a point in polar coordinates, e.g., ((r, \theta + 2\pi n)).

Important Concepts

• Pole: The origin in polar coordinates.
• Converting between Cartesian and Polar:
  • Polar to Cartesian: (x = r \cos \theta, y = r \sin \theta)
  • Cartesian to Polar: (r = \sqrt{x^2 + y^2}, \theta = \tan^{-1}(y/x))

Converting Between Coordinate Systems

Example Conversions

1. Polar to Cartesian:
   • ((-4, \frac{2\pi}{3})) converts to ((2, -2\sqrt{3}))
2. Cartesian to Polar:
   • ((-1, -1)) converts to ((\sqrt{2}, \frac{5\pi}{4}))

Converting Equations

• Example: Convert (2x - 5x^3 = 1 + xy) to polar coordinates:
  • Substitute (x = r\cos \theta, y = r\sin \theta)
  • Result: (2r\cos \theta - 5r^3\cos^3\theta = 1 + r^2\cos \theta\sin \theta)
• Example: Convert (r = -8\cos \theta)
  • Result: (x^2 + y^2 = -8x)

Common Polar Coordinate Graphs

Lines

1. (\theta = \beta) is a line through the origin, (y = \tan \beta x)
2. (r\cos \theta = a) is a vertical line, (x = a)
3. (r\sin \theta = b) is a horizontal line, (y = b)

Circles

1. Centered at Origin: (r = a)
2. Centered on x-axis: (r = 2a\cos \theta)
3. Centered on y-axis: (r = 2b\sin \theta)
4. General Circle Equation: (r = 2a\cos \theta + 2b\sin \theta)

Cardioids and Limacons

• Cardioids: (r = a \pm a\cos \theta) or (r = a \pm a\sin \theta)
• Limacons with Inner Loop: (r = a \pm b\cos \theta) or (r = a \pm b\sin \theta), (a < b)
• Limacons without Inner Loop: (r = a \pm b\cos \theta) or (r = a \pm b\sin \theta), (a > b)

Analyzing Graphs

• Determine when graphs pass through the origin by setting (r = 0).
• Example: For (r = 2 + 4\cos \theta), the graph passes through the origin at (\theta = \frac{2\pi}{3}, \frac{4\pi}{3}).